Relationship between zeros and coefficients of a cubic polynomial
If α, β, and γ are the zeros of ax^3+bx^2+cx+d, then α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a.
Practice This ConceptMain explanation
Teacher explanation
This result extends the quadratic relationship to cubics. It is useful when one or more zeros are known, or when a cubic polynomial is formed from given roots. The signs must be read carefully from the standard form. This relation saves time and is often used in higher-level algebra questions.
Example
For x^3-6x^2+11x-6, sum of zeros is 6, sum of pairwise products is 11, and product of zeros is 6.
Simple analogy
Cubic: sum, pair-sum, product; watch the signs.
Common confusion
Students often forget that the product of three zeros has a negative sign with d/a in the standard form ax^3+bx^2+cx+d.
Exam tip
Write the cubic in standard form first, then apply all three relations one by one.
Answer writing and exam use
1-mark use
Write the exact meaning of relationship between zeros and coefficients of a cubic polynomial in one clean line.
2-mark use
Define relationship between zeros and coefficients of a cubic polynomial and add one example or condition.
3-mark use
Explain relationship between zeros and coefficients of a cubic polynomial, show the method or example, and mention the common mistake.
Practice this concept with focused MCQs
Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.
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